Dimension-free bounds and structural results in communication complexity

Abstract

The goal of this thesis is to continue to build the bridge between communication complexity and analysis. More specifically, the purpose is to initiate a systematic study of dimension-free relations between basic communication complexity and query complexity measures and various matrix norms. In other words, our goal is to establish qualitative equivalences between complexity measures, namely to bound a measure solely as a function of another measure. This is in contrast to the more common framework in communication complexity where quantitative equivalences are the main focus of study and poly-logarithmic dependencies on the number of input bits are tolerated.Dimension-free bounds are closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We restate and propose several conjectures in this nature such as: Does every matrix with small randomized communication complexity contain a large all-zero or all-one submatrix [CLV19]? Does every Boolean function with small approximate Fourier algebra norm have large affine subspace on which the function is constant?We consider such questions for several communication and query complexity measures as well as various matrix and operator norms. In several cases, we achieve satisfying answers, while for some cases we show that such bounds do not exist.We establish that, in addition to applications in complexity theory, these problems arise naturally in operator theory and Harmonic analysis. We show that these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen's celebrated idempotent theorem regarding the Fourier algebra. i ContributionWork included The novel parts of this thesis are Chapters 4, 5, 6 and 7. These are based on the following joint work. Section 4.6 is not included in any manuscript.

Randomized communication and implicit graph representations

“…We write D Eq (M ) for the minimum cost of a 2-way deterministic protocol with Equality oracles. For computing adjacency in monotone graph classes (closed under edge & vertex deletions), all constant-cost randomized protocols can be put in this form [EHK22], but in general they cannot [HHH22b,HWZ22]. [HHP + 22] showed that D Eq (G) = O(1) implies that G has bounded sign-rank; our results explore the converse.…”

“…In the other direction, there are communication problems in BPP[1] that do not belong to D Eq [1], which was proved independently in [HHH22b] and [HWZ22], but the example in both cases, the 1-Hamming Distance problem (adjacency in the hypercube), is believed not to belong to UPP[1] [HHP + 22], which is implied by a positive answer to Question 1.6. In Section 5, we give two explicit examples (K 2,2 -free point-box incidences, and point-line incidences) in UPP[1] that do not belong to D Eq [1], which could possibly provide a negative answer to Question 1.6 if they belong to BPP[1], but point-line incidences are conjectured not to belong to BPP[1] in [CHHS23].…”

“…The study of constant-cost randomized communication was initiated independently in [HHH22b,HWZ22]. One motivation of [HWZ22] was that constant-cost communication is a special case of a well-studied open problem in structural graph theory and distributed computing, which asks to characterize the hereditary graph classes G that admit implicit representations (see e.g.…”

“…In the communication complexity literature, QS(Eq) was recently named the set of blocky matrices [HHH22b,AY22]. In graph theory, QS(Eq) are the adjacency matrices of disjoint unions of bicliques, also called bipartite equivalence graphs.…”

“…In the special case when B is the set of identity matrices, this definition appeared independently in [HHH22b,HWZ22] and subsequently in [EHK22], and the minimum t such that the above conditions hold is a "functional" analog of rank, recently called the functional blocky-rank in [AY22]. It is not difficult to show that this structural definition of constant-cost reductions is equivalent to the algorithmic one.…”